KAKUTANI'S FIXED POINT THEOREM AND THE MINIMAX THEOREM IN GAME THEORY YOUNGGEUN YOO Abstract. The minimax theorem is one of the most important results in game theory. It was first introduced by John von Neumann in the paper Zur Theorie Der Gesellschaftsspiele. Later, John Forbes Nash Jr. provided an alternative proof of the minimax theorem using Brouwer's fixed point theo- rem. We describe in detail Kakutani's proof of the minimax theorem using Kakutani's fixed point theorem, and discuss applications of Kakutani's fixed point theorem to economics and the theory of zero-sum games. Contents 1. Introduction 1 2. Kakutani's Fixed Point Theorem 3 3. The Minimax Theorem in Zero-Sum Games 6 Acknowledgments 8 References 9 1. Introduction Game theory predicts how rational individuals would behave under interdepen- dence. It provides a great mathematical tool to infer outcomes when people have conflicts of interests, which often happens in economic situations. The minimax theorem is one of the most important results in game theory. The theorem shows that there exists a strategy that both minimizes the maximum loss and maximizes the minimum gain. In other words, there is a strategy which rational people would take assuming the worst case scenario. Moreover, by working backwards from the outcome of the minimax theorem, it is possible to calculate the best strategy the players could take. In 1928, John von Neumann proved the minimax theorem using a notion of integral in Euclidean spaces. John Nash later provided an alternative proof of the minimax theorem using Brouwer's fixed point theorem. This paper aims to introduce Kakutani's fixed point theorem, a generalized version of Brouwer's fixed point theorem, and use it to provide an alternative proof of the minimax theorem. Then, we will discuss applications of Kakutani's fixed point theorem to economics and the theory of zero-sum games. Since this paper aims to be as self-contained as possible, the following are some basic topological concepts necessary for discussion and an introduction of Brouwer's fixed point theorem. Date: July 16 2016. 1 2 YOUNGGEUN YOO Definition 1.1. A convex hull of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X. Definition 1.2. A polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope. Definition 1.3. A simplex is a k-dimensional space polytope which is the convex k hull of its k + 1 vertices. Suppose the k + 1 points are u0; u1; :::; uk 2 R and are affinely independent, which means u1 − u0; u2 − u0; :::; uk − u0 are linearly independent. Then, the simplex determined by them is the set of points k X (1.4) C = fθ0u0 + θ1u1 + ::: + θkuk; θi ≥ 0 (i = 0; 1; 2; :::; k); θi = 1g i=0 Definition 1.5. A continuous mapping is such a function that maps convergent sequences into convergent sequences: if xn ! x then g(xn) ! g(x). Theorem 1.6 (Brouwer Fixed Point). If x ! '(x) is a continuous point-to-point mapping of an r-dimensional closed simplex S into itself, then there exists an x0 2 S such that x0 = '(x0): The proof of Brouwer's fixed point theorem uses the following Proposition 1.7. There exists no retraction D2 ! @D2 = S1. Moreover, there exists no retraction Dn+1 ! =partialDn+1 = Sn. The modern proof of this proposition is one line and uses the fundamental group. As familiarity with the fundamental group is not assumed, we take the proposition for granted and proceed to the proof of Brouwer's fixed point theorem. A reference for the proof is [4]. S1 D2 r(x) x f(x) Figure 1. Proof of Brouwer's fixed point theorem (Theorem 1.6). Start with two-dimensional case. Let x ! '(x) be a continuous mapping such that ' : D2 ! D2, where D2 is an unit disk. Let S1 be the boundary of D2. Assume '(x) 6= x for all x. Define r(x) 2 S1 be the intersection of S1 with the ray that starts from f(x) and goes through x (c.f. Figure 1). The map r(x) is continuous mapping from D2 to S1, and if x 2 S1 then r(x) = x. However, note that if j : S1 ! D2, then there exists no continuous map i : D2 ! S1 such that i ◦ j = id. If we only consider x 2 S1, KAKUTANI'S FIXED POINT THEOREM AND THE MINIMAX THEOREM IN GAME THEORY3 then '(x) can be considered as a mapping from S1 to D2. However, ' ◦ r(x) = id for x 2 S1; i.e. ' is a retraction D2 ! S1. Since this contradicts Proposition 1.7, it must be that '(x) = x for some x 2 D2. The proof for arbitrary dimension n is analogous, with D2 changed to Dn and S1 to Sn−1. (The proof that there exists n n n−1 no retraction D ! @D = S uses the higher homotopy group πn−1 in lieu of π1.) 2. Kakutani's Fixed Point Theorem Kakutani's fixed point theorem generalizes Brouwer's fixed point theorem in two aspects. A point-to-point mapping is generalized to point-to-set mapping, and continuous mapping is generalized to upper semi-continuous mapping. Definition 2.1. A point-to-set map is a relation where every input is associated with at least one output in which at least one input is associated with two or more outputs. Definition 2.2. A point-to-set mapping x ! Φ(x) 2 Ω(S) of S into Ω(S) is called upper semi-continuous if xn ! x0, yn 2 Φ(xn) and yn ! y0 imply y0 2 Φ(x0). Definition 2.3. Barycentric simplicial subdivision is a canonical algorithm to divide an arbitrary convex polytope into simplices with the same dimension, by connecting the barycenters of their faces. Lemma 2.1. Every bounded sequence of real numbers has a convergent subse- quence. Proof. Let (xn) be a sequence of real numbers bounded by [0; 1]. For sequences that are bounded by [a; b] where a; b 2 R, subtract a and divide by (b − a)=2 for each element of the sequence to obtain a sequence (xn), which converges if and only if the original sequence converges. Divide [0; 1] into two intervals [0; 1=2], [1=2; 1]. At least one of two intervals must contain infinitely many elements of (xn), since if both intervals contain finite number of elements then the union of both sets cannot be infinite. Repeatedly split the intervals into halves and let In (n = 1; 2; 3; :::) be the intervals that contain infinitely many elements in the each step. Then, n I1 ⊂ I2 ⊂ I3 ⊂ :::, where the length of In = (1=2) . Now, let the sequence (zn) be a subsequence of (xn) made up of one element from each interval. Then (zn) N converges since for any N 2 N, there exists m and n such that j(zm −zn)j < (1=2) N N since any interval In for n > N would have interval smaller than (1=2) , and (1=2) can get arbitrarily small so (zn) is a Cauchy sequence. Therefore, the subsequence (zn) converges. Lemma 2.2 (Bolzano-Weierstrass). Every bounded sequence in Rn has a conver- gent subsequence. n Proof. Let (xm) be a bounded sequence in R . Now consider the sequence (π1(xm)). This sequence is a bounded sequence of real numbers, so it has a convergent sub- sequence (π1(xmk )). Similarly, the sequences (π2(xm)), (π3(xm)); :::; (πn(xm)) all have convergent subsequence (π2(xmk )), (π3(xmk )); :::; (πn(xmk )). Let (zm) be a subsequence of (xm) such that (zm) = (π1(xmk ); π2(xmk ); :::; πn(xmk )). Since n- components of (zm) converges, (zm) converges as well. Corollary 2.2.1. Every sequence in a closed and bounded set S in Rn has a con- vergent subsequence which converges to a point in S. 4 YOUNGGEUN YOO Proof. Every sequence in a closed and bounded set is bounded, so it has convergent subsequence by Lemma 2.2. Since the set S is closed, such subsequence converges to a point in S. Theorem 2.4. Kakutani's Fixed Point Theorem 1: If x ! Φ(x) is an upper semi- continuous point-to-set mapping of an r-dimensional closed simplex S into Ω(S), then there exists an x0 2 S such that x0 2 Φ(x0). Proof. Let Sn be the nth barycentric simplicial subdivision of S. The natural number n denotes the number of barycentric simplicial subdivisions that are done. The examples of S0;S1, and S2 are Figure 2. Note that we only need the simplex S to be divided into smaller simplices, not necessarily through barycentric subdivision. Such subdivision was introduced as one of many possible subdivisions. 1 12 11 S = S0 S1 S2 Figure 2. Since S is an r-dimensional closed simplex, there are many smaller r-dimensional closed simplices in Sn. Choose one of those smaller simplices, which would have r + 1 vertices, and denote it Sx. Denote each vertex as x0; x1; :::; xr. Then, all the points in Sx can be represented by the equation θ0x0 +θ1x1 +···+θrxr; θi ≥ 0 (i = Pr 0; 1; 2; : : : ; r); i=0 θi = 1 by the definition of simplex. For each of those vertex xi, Φ(xi) would be a set of numbers since Φ(xi) is a point-to-set mapping. Let yi be an arbitrary point in Φ(xi). Let the simplex formed by y0; y1; :::; yr be Sy.